44 AMP: My experience is that what you said might not be true. I've traveled to Illinois with firearms and used them in competitions there, along with many other out-of-staters (none of whom had a FOID). I have a hunch that the competition organizers would adress this issue if the exemption wasn't real. As I stated earlier, multiple retailers would also sell ammo to me with an out-of-state driver's license.

The exception lies within the bit of code that NavyLT posted earlier:
(430 ILCS 65/2) (from Ch. 38, par. 83‑2)
Sec. 2. Firearm Owner's Identification Card required; exceptions.
(a) (1) No person may acquire or possess any firearm, stun gun, or taser within this State without having in his or her possession a Firearm Owner's Identification Card previously issued in his or her name by the Department of State Police under the provisions of this Act.

(b) The provisions of this Section regarding the possession of firearms, firearm ammunition, stun guns, and tasers do not apply to:

(9) Nonresidents whose firearms are unloaded and enclosed in a case; (or)(10) Nonresidents who are currently licensed or registered to possess a firearm in their resident state;

It says "licensed or registered." Right or wrong, I (and others) have taken that to mean "eligible" in cases where licensing or registration is not required in the respective states of residence. Perhaps some would disagree - it's certainly arguable.

If a non-resident is coming to Illinois to hunt and would like to bring their firearm, how do they legally transport it?
Non- residents must be legally eligible to possess or acquire firearms and ammunition in their state of residence.